Stone M., Goldbart P. Mathematics for Physics: A Guided Tour for Graduate Students

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11.4 Physical applications 395

Exercise 11.8: Use the infinitesimal homotopy relation to show that L and d commute,

i.e. for ω a p-form, we have

(

)

= L

(dω).

11.4 Physical applications

11.4.1 Maxwell’s equations

In relativistic

four-dimensional tensor notation the two source-free Maxwell’s equations

curl E =−

∂B

∂t

div B = 0, (11.66)

reduce to the single equation

∂F

µν

∂x

∂F

νλ

∂x

∂F

λµ

∂x

= 0, (11.67)

where

µν

⎛

⎜

⎝

0 −E

−E

0 B

−B

0 B

−B

⎞

⎟

⎠

. (11.68)

The “F” is traditional, for Michael Faraday. In form language, the relativistic equation

becomes the even more compact expression dF = 0, where

F ≡

µν

= B

dydz + B

dzdx + B

dxdy + E

dxdt + E

dydt + E

dzdt, (11.69)

is a Minkowski-space 2-form.

Exercise 11.9: Verify that the source-free Maxwell equations are indeed equivalent

to dF = 0.

The equation dF = 0 is automatically satisfied if we introduce a 4-vector 1-form potential

A =−φdt + A

dx + A

dy + A

dz and set F = dA.

In this section we will use units in which c = 

= µ

= 1. We take the Minkowski metric to be

µν

= diag (−1, 1, 1, 1) where x

= t, x

= x, etc.

396 11 Differential calculus on manifolds

The two Maxwell equations with sources

div D = ρ,

curl H = j +

∂D

∂t

, (11.70)

reduce in 4-tensor notation to the single equation

∂

µν

= J

. (11.71)

Here J

= (ρ, j) is the current 4-vector.

This source equation takes a little more work to express in form language, but it can

be done. We need a new concept: the Hodge “star” dual of a form. In d dimensions

the “” map takes a p-form to a (d − p)-form. It depends on both the metric and the

orientation. The latter means a canonical choice of the order in which to write our basis

forms, with orderings that differ by an even permutation being counted as the same. The

full d-dimensional definition involves the Levi-Civita duality operation of Chapter 10,

combined with the use of the metric tensor to raise indices. Recall that

√

g =



det g

µν

(In Minkowski-signature metrics we should replace

√

g by

√

−g.) We define “”tobe

a linear map

 :

∗

M ) →

(d−p)

∗

M ) (11.72)

such that

 dx

...dx

def

(d − p)!

√

...g



···j

p+1

···j

p+1

...dx

. (11.73)

Although this definition looks a trifle involved, computations involving it are not so

intimidating. The trick is to work, whenever possible, with oriented orthonormal frames.

If we are in Euclidean space and {e

∗i

, e

∗i

, ..., e

∗i

} is an ordering of the orthonormal

basis for (T

∗

M )

whose orientation is equivalent to {e

∗1

, e

∗2

, ..., e

∗d

} then

(e

∗i

∧ e

∗i

∧···∧e

∗i

) = e

∗i

p+1

∧ e

∗i

p+2

∧···∧e

∗i

. (11.74)

For example, in three dimensions, and with x, y, z our usual cartesian coordinates,

we have

 dx = dydz,

 dy = dzdx,

 dz = dxdy. (11.75)

11.4 Physical applications 397

An analogous method works for Minkowski-signature (−, +, +, +) metrics, except that

now we must include a minus sign for each negatively normed dt factor in the form

being “starred”. Taking {dt, dx, dy, dz} as our oriented basis, we therefore find

 dxdy =−dzdt,

 dydz =−dxdt,

 dzdx =−dydt,

 dxdt = dydz,

 dydt = dzdx,

 dzdt = dxdy. (11.76)

For example, the first of these equations is derived by observing that (dxdy)(−dzdt) =

dtdxdydz, and that there is no “dt” in the product dxdy. The fourth follows from observing

that (dxdt)(−dydx) = dtdxdydz, but there is a negative-normed “dt” in the product dxdt.

The  map is constructed so that if

α =

...i

···dx

, (11.77)

and

β =

...i

···dx

, (11.78)

then

α ∧(β) = β ∧ (α) =α, βσ , (11.79)

where the inner product α, β is defined to be the invariant

α, β=

···g

...i

...j

, (11.80)

and σ is the volume form

σ =

√

gdx

···dx

. (11.81)

In future we will write αβfor α ∧ (β). Bear in mind that the “” in this expression

is acting on β and is not some new kind of binary operation.

We now apply these ideas to Maxwell. From the field-strength 2-form

F = B

dydz + B

dzdx + B

dxdy + E

dxdt + E

dydt + E

dzdt, (11.82)

See for example: C. W. Misner, K. S. Thorn and J. A. Wheeler, Gravitation (MTW) p. 108.

398 11 Differential calculus on manifolds

we get a dual 2-form

F =−B

dxdt − B

dydt − B

dzdt + E

dydz + E

dzdx + E

dxdy . (11.83)

We can check that we have correctly computed the Hodge star of F by taking the wedge

product, for which we find

F  F =

µν

)σ = (B

+ B

− E

)dtdxdydz. (11.84)

Observe that the expression B

− E

is a Lorentz scalar. Similarly, from the current

1-form

J ≡ J

=−ρ dt + j

dx + j

dy + j

dz, (11.85)

we derive the dual current 3-form

J = ρ dxdydz −j

dtdydz − j

dtdzdx − j

dtdxdy , (11.86)

and check that

J  J = (J

)σ = (−ρ

+ j

)dtdxdydz. (11.87)

Observe that

d  J =



∂ρ

∂t

+ div j



dtdxdydz = 0 (11.88)

expresses the charge conservation law.

Writing out the terms explicitly shows that the source-containing Maxwell equations

reduce to d  F = J . All four Maxwell equations are therefore very compactly

expressed as

dF = 0, d  F = J .

Observe that current conservation d  J = 0 follows from the second Maxwell equation

as a consequence of d

= 0.

Exercise 11.10: Show that for a p-form ω in d Euclidean dimensions we have

ω= (−1)

p(d−p)

ω.

Show, further, that for a Minkowski metric an additional minus sign has to be inserted.

(For example, F =−F, even though (−1)

2(4−2)

=+1.)

11.4 Physical applications 399

11.4.2 Hamilton’s equations

Hamiltonian dynamics take place in phase space, a manifold with coordinates

, ..., q

, p

, ..., p

). Since momentum is a naturally covariant vector,

phase space

is usually the cotangent bundle T

∗

M of the configuration manifold M. We are writing

the indices on the p’s upstairs though, because we are considering them as coordinates

in T

∗

M .

We expect that you are familiar with Hamilton’s equations in their q, p setting. Here,

we shall describe them as they appear in a modern book on Mechanics, such asAbrahams

and Marsden’s Foundations of Mechanics, or V. I. Arnold’s Mathematical Methods of

Classical Mechanics.

Phase space is an example of a symplectic manifold, a manifold equipped with a

symplectic form – a closed, non-degenerate, 2-form field

ω =

. (11.89)

Recall that the word closed means that dω = 0. Non-degenerate means that for any

point x the statement that ω(X , Y ) = 0 for all vectors Y ∈ TM

implies that X = 0at

that point (or equivalently that for all x the matrix ω

(x) has an inverse ω

(x)).

Given a Hamiltonian function H on our symplectic manifold, we define a velocity

vector-field v

by solving

dH =−i

ω =−ω(v

, ) (11.90)

for v

. If the symplectic form is ω = dp

+ dp

+···+dp

, this is nothing

but a fancy form of Hamilton’s equations. To see this, we write

dH =

∂H

∂q

∂H

∂p

(11.91)

and use the customary notation (˙q

, ˙p

) for the velocity-in-phase-space components,

so that

=˙q

∂

∂q

+˙p

∂

∂p

. (11.92)

Now we work out

ω = dp

(˙q

∂

+˙p

∂

, )

=˙p

−˙q

, (11.93)

To convince yourself of this, remember that in quantum mechanics ˆp

=−i

∂

∂x

, and the gradient of a

function is a covector.

400 11 Differential calculus on manifolds

so, comparing coefficients of dp

and dq

on the two sides of dH =−i

ω, we read off

˙q

∂H

∂p

, ˙p

=−

∂H

∂q

. (11.94)

Darboux’ theorem, which we will not try to prove, says that for any point x we

can always find coordinates p, q, valid in some neighbourhood of x , such that ω =

+dp

+···dp

. Given this fact, it is not unreasonable to think that there

is little to be gained by using the abstract differential-form language. In simple cases this

is so, and the traditional methods are fine. It may be, however, that the neighbourhood

of x where the Darboux coordinates work is not the entire phase space, and we need to

cover the space with overlapping p, q coordinate charts. Then, what is a p in one chart

will usually be a combination of p’s and q’s in another. In this case, the traditional form of

Hamilton’s equations loses its appeal in comparison to the coordinate-free dH =−i

ω.

Given two functions H

, H

we can define their Poisson bracket {H

, H

}. Its impor-

tance lies in Dirac’s observation that the passage from classical mechanics to quantum

mechanics is accomplished by replacing the Poisson bracket of two quantities, A and B,

with the commutator of the corresponding operators

A, and

B]←→{A, B}+O





. (11.95)

We define the Poisson bracket by

, H

}

def

= v

. (11.96)

Now, v

= dH

), and Hamilton’s equations say that dH

) =

ω(v

, v

). Thus,

, H

}=ω(v

, v

). (11.97)

The skew symmetry of ω(v

, v

) shows that despite the asymmetrical appearance of

the definition we have skew symmetry: {H

, H

}=−{H

, H

Moreover, since

) = (v

+ H

), (11.98)

the Poisson bracket is a derivation:

, H

}={H

, H

+ H

, H

}. (11.99)

Our definition differs in sign from the traditional one, but has the advantage of minimizing the number of

minus signs in subsequent equations.

11.4 Physical applications 401

Neither the skew symmetry nor the derivation property require the condition that dω = 0.

What does need ω to be closed is the Jacobi identity:

{{H

, H

}, H

}+{{H

, H

}, H

}+{{H

, H

}, H

}=0. (11.100)

We establish Jacobi by using Cartan’s formula in the form

dω(v

, v

) = v

ω(v

, v

) + v

ω(v

, v

) + v

ω(v

, v

)

− ω([v

, v

], v

) − ω([v

, v

], v

)

− ω([v

, v

], v

). (11.101)

It is relatively straightforward to interpret each term in the first of these lines as Poisson

brackets. For example,

ω(v

, v

) = v

, H

}={H

, {H

, H

}}. (11.102)

Relating the terms in the second line to Poisson brackets requires a little more effort. We

proceed as follows:

ω([v

, v

], v

) =−ω(v

, [v

, v

])

= dH

([v

, v

])

=[v

, v

= v

) − v

)

={H

, {H

, H

}}−{H

, {H

, H

}}

={H

, {H

, H

}}+{H

, {H

, H

}}. (11.103)

Adding everything together now shows that

0 = dω(v

, v

)

= −{{H

, H

}, H

}−{{H

, H

}, H

}−{{H

, H

}, H

}. (11.104)

If we rearrange the Jacobi identity as

, {H

, H

}}−{H

, {H

, H

}}={{H

, H

}, H

}, (11.105)

we see that it is equivalent to

, v

]=v

}

The algebra of Poisson brackets is therefore homomorphic to the algebra of the Lie

brackets. The correspondence is not an isomorphism, however: the assignment H (→ v

fails to be one-to-one because constant functions map to the zero vector field.

402 11 Differential calculus on manifolds

Exercise 11.11: Use the infinitesimal homotopy relation, to show that L

ω = 0, where

is the vector field corresponding to H . Suppose now that the phase space is 2n

dimensional. Show that in local Darboux coordinates the 2n-form ω

/n! is, up to a sign,

the phase-space volume element d

q. Show that L

/n!=0 and that this result

is Liouville’s theorem on the conservation of phase-space volume.

The classical mechanics of spin

It is sometimes said in books on quantum mechanics that the spin of an electron, or other

elementary particle, is a purely quantum concept and cannot be described by classical

mechanics. This statement is false, but spin is the simplest system in which traditional

physicists’ methods become ugly and it helps to use the modern symplectic language. A

“spin” S can be regarded as a fixed length vector that can point in any direction in R

We will take it to be of unit length so that its components are

= sin θ cos φ,

= sin θ sin φ,

= cos θ, (11.106)

where θ and φ are polar coordinates on the 2-sphere S

The surface of the sphere turns out to be both the configuration space and the phase

space. In particular the phase space for a spin is not the cotangent bundle of the configu-

ration space. This has to be so: we learned from Niels Bohr that a 2n-dimensional phase

space contains roughly one quantum state for every 

of phase-space volume. A cotan-

gent bundle always has infinite volume, so its corresponding Hilbert space is necessarily

infinite dimensional. A quantum spin, however, has a finite-dimensional Hilbert space

so its classical phase space must have a finite total volume. This finite-volume phase

space seems unnatural in the traditional view of mechanics, but it fits comfortably into

the modern symplectic picture.

We want to treat all points on the sphere alike, and so it is natural to take the symplectic

2-form to be proportional to the element of area. Suppose that ω = sin θ dθdφ. We could

write ω =−d cos θ dφ and regard φ as “q” and −cos θ as “p” (Darboux’ theorem in

action!), but this identification is singular at the north and south poles of the sphere, and,

besides, it obscures the spherical symmetry of the problem, which is manifest when we

think of ω as d(area).

Let us take our Hamiltonian to be H = BS

, corresponding to an applied magnetic

field in the x-direction, and see what Hamilton’s equations give for the motion. First we

take the exterior derivative

d(BS

) = B(cos θ cos φdθ − sin θ sin φdφ). (11.107)

This is to be set equal to

−ω(v

, ) = v

(−sin θ)dφ +v

sin θdθ . (11.108)

11.5 Covariant derivatives 403

Comparing coefficients of dθ and dφ,weget

(BS

)

= v

∂

+ v

∂

= B(sin φ∂

+ cos φ cot θ∂

), (11.109)

i.e. B times the velocity vector for a rotation about the x-axis. This velocity field therefore

describes a steady Larmor precession of the spin about the applied field. This is exactly

the motion predicted by quantum mechanics. Similarly, setting B = 1, we find

=−cos φ∂

+ sin φ cot θ∂

=−∂

. (11.110)

From these velocity fields we can compute the Poisson brackets:

, S

}=ω(v

, v

)

= sin θdθ dφ(sin φ∂

+ cos φ cot θ∂

, −cos φ∂

+ sin φ cot θ∂

)

= sin θ(sin

φ cot θ + cos

φ cot θ)

= cos θ = S

Repeating the exercise leads to

, S

}=S

, S

}=S

, S

}=S

. (11.111)

These Poisson brackets for our classical “spin” are to be compared with the commutation

relations [

]=i

etc. for the quantum spin operators

11.5 Covariant derivatives

Covariant derivatives are a general class of derivatives that act on sections of a vector or

tensor bundle over a manifold. We will begin by considering derivatives on the tangent

bundle, and in the exercises indicate how the idea generalizes to other bundles.

11.5.1 Connections

The Lie and exterior derivatives require no structure beyond that which comes for

free with our manifold. Another type of derivative that can act on tangent-space vec-

tors and tensors is the covariant derivative ∇

≡ X

∇

. This requires an additional

mathematical object called an affine connection.

The covariant derivative is defined by:

(i) Its action on scalar functions as

∇

f = Xf . (11.112)

404 11 Differential calculus on manifolds

(ii) Its action on a basis set of tangent-vector fields e

(x) = e

(x)∂

(a local frame, or

vielbein

) by introducing a set of functions ω

(x) and setting

∇

= e

. (11.113)

(iii) Extending this definition to any other type of tensor by requiring ∇

to be a

derivation.

(iv) Requiring that the result of applying ∇

to a tensor is a tensor of the same type.

The set of functions ω

(x) is the connection. In any local coordinate chart we can choose

them at will, and different choices define different covariant derivatives. (There may be

global compatibility constraints, however, which appear when we assemble the charts

into an atlas.)

Warning: Despite having the appearance of one, ω

is not a tensor. It transforms

inhomogeneously under a change of frame or coordinates – see Equation (11.132).

We can, of course, take as our basis vectors the coordinate vectors e

≡ ∂

. When we

do this it is traditional to use the symbol  for the coordinate frame connection instead

of ω. Thus,

∇

≡∇

= e



νµ

. (11.114)

The numbers 

νµ

are often called Christoffel symbols.

As an example consider the covariant derivative of a vector f

. Using the derivation

property we have

∇

) = (∂

+ f

∇

= (∂

+ f



νµ

= e



∂

+ f



λµ



. (11.115)

In the first line we have used the defining property that ∇

acts on the functions f

∂

, and in the last line we interchanged the dummy indices ν and λ. We often abuse the

notation by writing only the components, and set

∇

= ∂

+ f



λµ

. (11.116)

Similarly, acting on the components of a mixed tensor, we would write

∇

βγ

= ∂

βγ

+ 

βγ

− 

βµ

λγ

− 

γµ

βλ

. (11.117)

When we use this notation, we are no longer regarding the tensor components as

“functions”.

In practice viel, “many”, is replaced by the appropriate German numeral: ein-, zwei-, drei-, vier-, fünf-, ...,

indicating the dimension. The word bein means “leg”.